Correlation of leaf heights at mesoscopic separation in the critical beta-splitting tree

In the critical beta-splitting random tree on $n$ leaves, in its interval-splitting form with the interval closed into a cycle, let $D^{(1)}_n$ and $D^{(2)}_n$ be the continuous heights of a uniformly random leaf and of the leaf $r$ places further round the cycle. We show that, uniformly in $1\le r\le n/2$, \[ \operatorname{Cov}\bigl(D^{(1)}_n,D^{(2)}_n\bigr)=\sigma^2\log(n/r)+O\bigl(\sqrt{\log n}\bigr),\qquad \operatorname{Var}(D_n)=\sigma^2\log n+O\bigl(\sqrt{\log n}\bigr), \] with $\sigma^2=2\zeta(3)/\zeta(2)^3$. Hence the correlation coefficient equals $1-\log r/\log n+O((\log n)^{-1/2})$, and it tends to $1-\alpha$ whenever $\log r_n/\log n\to\alpha\in[0,1]$. This answers Open Problem 1 of Aldous and Janson. The proof follows a positional potential of each leaf, which decreases at rate $\zeta(2)$ up to errors that are square integrable along the whole path, and compares the two martingales it produces through their bracket up to the separation of the two leaves.